Mapping Cone Connections and their Yang-Mills Functional

TL;DR

This paper introduces the cone Yang-Mills functional for pairs of connections and scalars derived from dimensionally reducing Yang-Mills theory, analyzing its solutions and their properties, including a duality condition and classification of flat solutions.

Contribution

It defines a new cone Yang-Mills functional from dimensional reduction, explores its critical solutions, and classifies principal bundles with cone-flat solutions.

Findings

Critical solutions satisfy a duality condition extending Bogomolny equations.

Zero solutions are classified algebraically for non-degenerate two-forms.

The functional generalizes Yang-Mills theory in a new geometric setting.

Abstract

For a given closed two-form, we introduce the cone Yang-Mills functional which is a Yang-Mills-type functional for a pair $(A,B)$, a connection one-form $A$ and a scalar $B$ taking value in the adjoint representation of a Lie group. The functional arises naturally from dimensionally reducing the Yang-Mills functional over the fiber of a circle bundle with the two-form being the Euler class. We write down the Euler-Lagrange equations of the functional and present some of the properties of its critical solutions, especially in comparison with Yang-Mills solutions. We show that a special class of three-dimensional solutions satisfy a duality condition which generalizes the Bogomolny monopole equations. Moreover, we analyze the zero solutions of the cone Yang-Mills functional and give an algebraic classification characterizing principal bundles that carry such cone-flat solutions when the two-form is non-degenerate.